Problem 24 · 2023 Math Kangaroo
Stretch
Geometry & Measurement
symmetry
A circle with midpoint \((75\,|\,30)\) and radius 10 is cut from a rectangle with vertices \((0\,|\,0)\), \((100\,|\,0)\), \((100\,|\,50)\) and \((0\,|\,50)\). What is the gradient of the straight line that goes through the point \((75\,|\,30)\) and divides the remaining part of the rectangle into two parts with equal area?
Show answer
Answer: A — \(\frac{1}{5}\)
Show hints
Hint 1 of 2
A line through the centre of a circle always halves that circle's area.
Still stuck? Show hint 2 →
Hint 2 of 2
So the line only needs to bisect the rectangle — which means passing through the rectangle's centre too.
Show solution
Approach: a center line bisects both shapes
- Any line through the hole's centre (75,30) splits the circular hole into two equal halves.
- To split the rest equally, the line must also bisect the rectangle, i.e. pass through its centre (50,25).
- The line through (75,30) and (50,25) has slope (30−25)/(75−50) = 5/25 = 1/5.
- So the gradient is 1/5.
Mark:
· log in to save